How many distinct digits can appear as the units digit of an integral perfect-square number?
Solution: Let $d$ be the last digit of a number $n$.  Then $n^2 \equiv d^2 \pmod{10}$, so the units digit of $n^2$ is the same as the units digit of $d^2$.  Checking all the digits from 0 to 9, we find that the possible units digits of $d^2$ are 0, 1, 4, 5, 6, and 9, for a total of $\boxed{6}$.